Updating Homebrew...
==> Downloading https://homebrew.bintray.com/bottles/mysql@5.6-5.6.37.el_capitan.bottle.tar.gz
Already downloaded: /Users/ryosaito/Library/Caches/Homebrew/mysql@5.6-5.6.37.el_capitan.bottle.tar.gz
==> Pouring mysql@5.6-5.6.37.el_capitan.bottle.tar.gz
==> /usr/local/Cellar/mysql@5.6/5.6.37/bin/mysql_install_db --verbose --user=ryosaito --basedir=/usr/local/Cellar/mysql@5.6/5.6.37 --datadir=/usr/local/var/mysql --tmpdir=/tmp
==> Caveats
A "/etc/my.cnf" from another install may interfere with a Homebrew-built
server starting up correctly.
MySQL is configured to only allow connections from localhost by default
To connect:
mysql -uroot
This formula is keg-only, which means it was not symlinked into /usr/local,
because this is an alternate version of another formula.
If you need to have this software first in your PATH run:
echo 'export PATH="/usr/local/opt/mysql@5.6/bin:$PATH"' >> ~/.bash_profile
For compilers to find this software you may need to set:
LDFLAGS: -L/usr/local/opt/mysql@5.6/lib
CPPFLAGS: -I/usr/local/opt/mysql@5.6/include
To have launchd start mysql@5.6 now and restart at login:
brew services start mysql@5.6
Or, if you don't want/need a background service you can just run:
/usr/local/opt/mysql@5.6/bin/mysql.server start
==> Summary
🍺 /usr/local/Cellar/mysql@5.6/5.6.37: 346 files, 154.8MB

出てきた説明を読んでみると、

If you need to have this software first in your PATH run:
echo ‘export PATH=“/usr/local/opt/mysql@5.6/bin:$PATH”’ >> ~/.bash_profile

Welcome to the MySQL monitor. Commands end with ; or \g.
Your MySQL connection id is 1
Server version: 5.6.37 Homebrew
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mysql>